Typical property of convex sets (Q1814466)

For any nonempty compact convex set \(U\subset \mathbb{R}^ n\) and any \(\phi\in \mathbb{R}^ n\) we set \(u(\phi)=\{u\in U: \langle u,\phi\rangle =\max_{v\in U}\langle v,\phi\rangle\}\). For any \(\phi: [a,b]\to \mathbb{R}^ n\) we denote by \(V(a,b,\phi)\) the set of all nonempty compact convex sets \(U\subset \mathbb{R}^ n\) such that the family \(U(\phi(t))\) has a selector of finite variation. Assume that \(n\geq 3\) and \(\phi\) is a smooth function such that \[ (d/dt)(\phi(t)/\| \phi(t)\|)_{\mid t=t_ 0}\neq 0 \] for some \(t_ 0\in (a,b)\). Then \(V(a,b,\phi)\) is an \(F_ \sigma\)-set of first category in the space of all convex compact sets. Some applications to linear optimal control problems are also given.